Optimal Tax Policy under Habit Formation and Capital
Utilization.
Goncalo Monteiro
State University of New York at Buffalo, NY 14260
Adam Cook
State University of New York at Buffalo, NY 14260
Sanjoy Dey
State University of New York at Buffalo, NY 14260
Abstract
Our objective is to investigate how the combination of habit formation with endogenous capital
utilization decisions affects the process of economic growth. We find that in the presence of
positive productivity growth, habit formation in consumption reduces the rate of capital utilization
while increasing the long run stock of capital. We show in the Chamley-Judd result of zero income
capital taxation is robust to the presence of endogenous capital utilization. The result even extends
to the short-run as long as there are no depreciation allowances, but breaks if those are present.
Our policy simulations show that steady state level of capital utilization is not affected by changes
in income tax, but its reduced on impact to accommodate for the constancy of capital stock Our
numerical simulations also show important differences in the time adjustment followed by key
variables, as a response to tax policy.
Keywords: habit formation; capital utilization; optimal taxation; Economic growth; user cost.
JEL classification: D91, E21, O41
1. Introduction
Recently, Solow(2005) advocated for the considerations of demand-side and supply-side
interaction in the study of economic growth. According to Solow:
“…some sort of endogenous knitting-together of the [short-run] fluctuations ad growth
contexts is needed, and not only for the sake of neatness: the short-run and its uncertainties
affect the long-run through the volume of investment … and the growth forces in the economy
probably influence the frequency and amplitude of short-run fluctuations.”
The standard Ramsey growth model is certainly at odds with this argument. On the
preference (demand-side) it assumes agent’s utility functions, at any point in time, depend only
upon contemporaneous variables such as current consumption (sometimes leisure). On the
production side (supply-side) it generally assumes production depend on physical capital
9sometimes labor or human capital) with full utilization of the installed capital. Therefore, in
order to address Solow’s suggestion we need to depart from the standard Ramsey model.
The current work, tries to account for the interaction between demand-side and supplyside forces by departing from the Ramsey model in two distinct ways. On the preference side we
recognize the limitations of this time-separable specification of utility have become increasingly
recognized, and thus allow current utility to depend not only on current consumption, but also on
past consumption levels, which provide a benchmark against which current consumption can be
assessed. This type of model are generally called models of habit-formation and were first
introduced by Ryder and Heal (1973), and recently have been applied to a variety of issues,
incuding but not limited to: asset pricing (Constantinides, 1990, Campbell and Cochrane, 1999),
consumption behavior (Osborn, 1988, Fearson and Constantinides, 1991, Dynan, 2000), shortrun macroeconomic stabilization (Ljundqvist and Uhlig, 2000, Fuhrer, 2000), exchange rate
behavior (Mansoorian, 1988) and economic growth (Carroll, Overland, and Weil, 1997, 2000,
Fuhrer Alvarez-Cuadrado, Monteiro, and Turnovsky, 2004, Alonso-Carrera, Caballé, Raurich,
2005, Turnovsky and Monteiro, 2007).
On the production side we use the Keynes’ (1936) idea that firms increase the intensity of
capital usage during a certain working period, and this leads to accelerated depreciation, thus
affecting future capital accumulation. Contrary to what happens with the habit formation, the
idea of higher wear and tear on equipment resulting from higher utilization rate, has received
little interest by economic growth theory contrasting with what happens in the business cycle
literature. An explanation for this behavior may be that while it seems natural to include capital
utilization as an optimal decision variable to explain cyclical variation in output, it is less evident
that the underutilization of capital would persist in the long-run, and thus play a key role in
explaining economic growth. . Dalgaard (2003), however, shows, non-negligible long-run
productivity differences can arise from plausible changes in the capital utilization rate, thus
making this inclusion in the economic growth relevant.
A few notable exceptions to the neglect mentioned above are: Licandro, Puch and RuizTamarit (2001), who analyze the equilibrium dynamics of an optimal growth model with
endogenous depreciation, variable capital utilization and maintenance costs and they conclude
that in the long-run capital is optimally underutilized and maintenance activities are optimally
undertaken. Rumbos and Auernheimer (2001), introduce a variable rate of capital utilization into
a modified Ramsey model, whereas Dalgaard (2003) introduces endogenous capital utilization
decisions into the standard neoclassical model. Both conclude that this reduces the speed of
convergence. Marquez and Ruiz-Tamarit (2004) extend the AK model by introducing adjustment
and maintenance costs. The most recent and “extensive” work belongs to Chatterjee (2005),
which accounts for the optimal choice of capital utilization in a growth model employing a
general production function (that accounts for the Ramsey, the AK and the semi-endogenous
growth models), and shows that capital utilization plays a crucial role slowing down the speed of
convergence.
In spite of the empirical support for capital utilization and habit formation, the
combination of these two concepts has only been done (to our knowledge) by Marc-André
Letendre(2004) in the very different context of a small open economy model calibrated to
replicate the Canadian economy. It is our view that combining this two ideas is in line with
Solow(2005) suggestion, but more importantly considering endogenous capital utilization
decisions under time separable utility may yield misleading conclusions if in fact preferences are
characterized by a high degree of complementarity between consumption at successive moments,
as the empirical evidence suggests [see Fuhrer (2000)]. On the other hand, it is also true that
considering habit formation in consumption, while ignoring capital utilization decisions, may
yield misleading conclusions if in fact the rate of capital utilization falls substantially below the
full utilization rate.
Of the studies cited, our analysis is closest to Chatterjee (2005)whose conclusion is that
incorporating capital utilization decision into the Ramsey model helps to resolve the discrepancy
between theory predictions and actual speed of convergence. Our paper objectives are a bit
broader than the speed of convergence debate; the goal is to investigate the effects combining
endogenous capital utilization with habit formation in consumption, on the predictions of the
Ramsey growth model. Therefore, we expand the Ramsey growth model to account for the
presence of a reference benchmark on consumption, as well as endogenous capital utilization
decisions. In particular, we consider the implications of habit formation on the optimal decisions
of capital utilization using a one sector neoclassical growth model in which labor experiences
positive productivity growth. In addition, we proceed to do some optimal tax policy analysis as
well as some numerical simulations of the effect of tax policy instruments. It is worth mentioning
that the model is general enough to allow for a comparison of our results not only with the
capital utilization model, but also with the model of habit formation and the traditional
neoclassical model. Therefore, we extend this comparison whenever it is relevant.
There are several key results that we wish to stress at the outset: the first and possibly
most general result is that the steady state of capital utilization and depreciation are only affected
by the weight of habits in the utility function, in the presence of labor productivity growth. In the
presence of labor productivity growth, the presence of habits raises (reduces) the steady state of
capital (rate of capital utilization) relative to the time separable preferences case, presented in
Chatterjee (2005). A possible explanation could be that agents do not like negative consumption
shocks, and their effect increases with the weight (  ) assigned to the consumption reference. To
guard against these shocks, a more addicted (higher gamma) agent will lower the rate of his
firm's capital utilization in order to lower the rate of capital depreciation and increase the pool of
available capital just in case he experiences a negative shock and must consume more of his
capital stock to maintain his habit.
Second, the introduction of endogenous capital utilization has implications for the
optimal tax analysis. An immediate implication is, we must distinguish between the
intertemporal (on capital accumulation) and the intratemporal (on capital utilization) effects of
optimal tax policy. The Chamley-Judd result of zero long-run optimal income holds even in the
presence of depreciation allowances. Third, the Chamley-Judd result fails to hold in the shortrun in the presence of capital depreciation allowances.
There are also some important results coming from the tax policy analysis. First, an
increase in the income tax rate reduces utilization in the short-run, but has no effect on the longrun. Second, increases in depreciation allowances increase the utilization rate by a larger
percentage in the short-run than in the long-run. The intuition is simple and is just a response to
the fact that capital cannot adjust in the short-run.
The paper is organized as follows. Section 2 lays out the model. Section 3 characterizes
the corresponding macroeconomic behavior for the decentralized economy and the cental
planner solution is presented in section 4. The optimal tax analysis is conducted in section 5 and
we analyze the effects of shocking the economy in section 6. Section 7 concludes, and an
appendix provides some technical detail.
2. Model
2.1 Preferences
Consider an economy populated by N identical and infinitely lived households that grow
at an exogenous rate N N  n . At any point in time, agents derive utility from current
consumption, Ci and a reference consumption level, H i . The agent’s objective is to maximize
the intertemporal iso-elastic utility function:
1

1 


0
1
 Ci 
 
 Hi 
e   t dt
  1;0    1
(1)
the condition 0    1 reflects the presence of non-satiation and this is guaranteed if an increase
in a uniformly maintained consumption level increases utility, i.e. UC  Ci , Ci   U H  Ci , Ci   0 .
In other words non-satiation is guaranteed if the reference level augments the direct effect of
consumption, or, if it is offsetting, it is dominated by the direct effect.
Following Ryder and Heal (1973) we assume the agent’s reference stock is an
exponentially declining weighted average of his own past levels of consumption, and can be
specified by
Hi  t     Ci e  t  d
t
 0

(2)
Differentiating (2) with respect to time yields the following rate of adjustment for the reference
stock,
Hi    Ci  Hi 
(3)
where  measures the relative importance of, or the weight assigned to, recent consumption in
determining the reference stock. Hence, higher values of  lead to lower level of persistence of
the reference benchmark in consumption.
2.2 Production
On the production side, agents make decisions at two distinct levels. First, and at any
instant in time, agents must choose the optimal level of investment, thus determining the flow of
capital into the economy. Second, given the level of capital chosen for the economy, agents
decide how much to use in production, i.e. it chooses the rate of utilization of the accumulated
stock. Therefore, individual output is determined by the flow of services derived from the
available level of capital in the economy at a certain time, K is , and the level of inelastically
supplied labor, Li . Assuming a Cobb-Douglas production individual output is determined by,
Yi    ALi 

K 
s 1
i
(4)
where K is  uK i represents the fractional flow of capital services used in final good production,
and u represents the rate of utilization or the intensity at which the total available stock of capital
in the economy is utilized1. In addition, it is also assumed labor productivity grows at the
exogenous growth rate, A A  g .
An immediate implication of introducing capital utilization in the model is that it no
longer makes sense to consider, as is traditionally the case, that we have a constant depreciation
rate. In other words, it is reasonable to consider that the rate of depreciation is increasing with
the rate of utilization, simply because the more intensively the agent utilizes the capital, (i.e. the
higher rate of utilization) the higher the wear and tear and thus the higher the rate of
depreciation. Therefore, the rate of depreciation will be endogenously determined in the model
and will be an increasing function of u, and can be specified as:
  u   du ,
  1, d  0, 0    u   1
(5)
where    u   0,    u   0 and   u   u    u  measures the elasticity of depreciation with
respect to the rate of capital utilization.
2.3 Government
Whereas in models with full utilization and constant capital depreciation there is no
significant difference between allowing the government to tax gross rather than net capital, the
same is not true in our model of capital utilization. This happens because endogenizing the capital
utilization decision introduces a new distortion on capital income taxation.
In models of full capital utilization, agents need only worry about the intertemporal
allocation decision, whereas in models of endogenous capital utilization decision agents must
also worry about a static decision of how much capital to use at each instant in time given the
stock of capital. In other words, by endogenizing capital utilization, capital supply is perfectly
elastic with respect to any tax rate both in the long run (dynamic intertemporal margin), as well
as the short-run (static intratemporal margin)
1
This definition follows Taubman and Wilkinson (1970), Calvo (1975) and Chatterjee (2005)
In light of this, we assume the government levies constant taxes on income,  ys , and
consumption ,  c , and permits capital depreciation allowances,  d . In addition the government
maintains a balanced budget, rebating all taxes revenues and allowances as lump sum transfers:
T   ys  r s uK  ws AL    cC   d   u  K
(6)
where r s is the rental rate of capital services, and w s is the effective real wage rate.
3. Macroeconomic Equilibrium: Decentralized case
Output can be used for consumption or investment. Thus, the individual in the decentralized
economy budget constraint is
K i  1   ys  r s uKi  ws ALi    n  1   d    u   K i  1   c  Ci  Ti
(7)
The individual chooses his consumption, the rate of capital utilization, the rate of capital
accumulation to maximize (1), subject to the production (4), to the rate of depreciation (5), the
accumulation equation (7). If we define c  Ci AL , k  Ki AL and h  H i AL , and let
U  c, h    ch 
1
1   , the agent’s optimization problem can be equivalently expressed as:

max  U  c, h  e t
(8)
0
s.t. k  1   ys  r s uk  ws    n  g  1   d    u   k  1   c  c  T
(9)
where        11    g  and to simplify the notation, we let U c and U h denote the partial
derivatives of U  c, h  with respect to c and h The optimality conditions for this problem are,
U c   1   c 
1    r  1    d u
s
y

s
(10a)
 1
d

 1   ys  r s u  1   d  du   n  g 

(10b)
(10c)
where  denotes the agent’s shadow value of capital, together with the transversality condition:
lim e t   t  k  t   0
t 
(10d)
The interpretation of the optimality conditions (10a)-(10c) is standards in growth
literature. Equation (10a) equates the utility of an additional unit of consumption to the tax
adjusted shadow value of capital taking into account that utility depends upon current
consumption relative to the benchmark; equation (10b) determines the optimal rate of capital
utilization by equating the marginal benefit to marginal cost (in a way this is an intratemporal or
static allocation condition). Equation (10c) is the standard intertemporal allocation condition
equating the marginal product of capital to the rate of return on consumption.
Using the normalized variable definition on the production function (4) and the
adjustment for the reference stock (3), yields:
y    Kis 
1
   uk 
1
(11)
h    c  h   gh
(12)
On the production side, the representative firm takes the effective real wage w s and the rental
rate of capital services r s as given, and rents capital and labor services from the representative
agent to maximize profits,  :
Max     ALi 

 Li , Ki 
K 
s 1
i
 r s K is  ws ALi
(13)
With the following first order conditions:
rs 
ws 

Yi


  1    ALi   K is    1    uk 
s
K i
1
Yi
 1
1
   ALi   Kis     uk 
ALi
(13a)
(13b)
Before we proceed to determine the macroeconomic equilibrium we should point that the
proposition presented in Chatterjee2 (2005) still holds here. This proposition states that, as long
as the rate of depreciation depends on the intensity at which the available stock of capital in the
2
See section 3.3. page 2101
economy is utilized  u  , agents will always find it optimal not to use the full amount of capital in
the economy, i.e. 0  u  1 .
We now proceed to derive the macroeconomic equilibrium. The first step is to replace
(13a) into (10b) and solve that for the rate of capital utilization as a function of efficient units of
capital and the tax rates, which yields,
1

 1   ys   1      1  

  1
  1

u k   
k
 Bk
 1   d  d  


(14)
and the depreciation function (5) becomes,
   
  u   d  Bk   1 



(15)
1
 1   ys   1      1

.
B
 1   d  d  


where
In addition, substituting (14) in the production function (11) specified in units of efficiency,
yields,
y  f  k    B k
with  
(16)
  11   
   1
Expressions (14)-(16) are equivalent to the results presented in Chatterjee (2005), if we
assume no government involvement, i.e. zero tax rates. This is not surprising, because the
reference stock is related to the preference side of the model and not to the production side, and
thus does not affect the use capital directly. These expressions show a positive (negative)
relationship between the marginal product of capital (aggregate capital stock) and the rate of
utilization3.
From equation (10.a) we obtain
3
or alternatively the rate of depreciation
c  C ( h,  )
(17a)
C  h,  
 2U
, with i  h,  , we have that
with i, j  c, h and Ci 
i
ij
Defining U ij 
Ch  
1   c   0,
U ch
; C 
U cc
U cc
(17b)
Using (17a) and (6) the macroeconomic equilibrium is terms of the canonical system is
summarized by expressions (9), (12), (10.c)
 1   ys  1    
 k  c  h,     n  g  k
k   B1 1 

 1   d  


(18a)
h    c(h,  )  h   gh
(18b)

   1  1  1 
B k 
  

(18c)
    g  n   1    1   ys  

(18c)
Imposing the stationary conditions, k  h    0 on the system (18a)-(18c) we can
determine the steady state values, for k , c, and h in the following recursive manner. First, (18c)
yields the solution for capital. Second, given k (18a) determines the solution for c . Third,
substituting this value of c into (18b) we get the steady state value for h [alternatively (18b) can
be used to yield the consumption-habit ratio]. Finally we use expression (10a) to determine the
solution for the shadow value of capital,  . Letting tildes denote the steady state values, we may
summarize these expressions as follows:
  1     1 1   ys  B1
k 
   g  n 

c B
1
h

g
1
 1


 1   ys  1    
1 
 k   n  g  k

 1   d  


c
c g


h
(19a)
(19b)
(19c)

 
U c c, h
(19d)
1c
In addition, using the steady state value of capital (16a) on (14) and (15) the steady state
values for the rate of capital utilization and the rate of depreciation are
1
1
   g  n          11     1 g  n  
u

 
1   d  d   1

 1   d  d   1 

(20a)
and
       11     1 g  n 

1   d   1


 
(20b)
The following observations about the steady state can be made: First, we see that in
contrast to the steady state level of capital which depends both upon the income tax rate,  ys 
and the rate of depreciation allowance  d  , the rate of capital utilization depends only on the rate
of depreciation allowances. Intuitively, the decision to accumulate capital depends on both the
amount of disposable income, as well as the intensity with which it will be used. The two
components are affected by different government policies. On the other hand, the decision about
the intensity of capital utilization is made taking capital as given, and thus is only affected by the
government depreciation allowances.
It can easily be shown that,
du
 0;
d d
dk
 0;
d d
dk
0
d ys
which intuitively can be explained because a rise in income taxation reduces the amount of
disposable income and thus reduces investment. The effect of the depreciation allowance on
utilization and capital is simple and intuitive to understand. Depreciation allowances lead agents
to increase capital utilization because of the tax savings (rebate) resulting from utilization and
thus the larger the rebate the larger the incentive to increase utilization. In contrast, and for the
exact opposite reason, the incentive to increase utilization detracts agents from capital
accumulation.
Second, in the absence of productivity growth,  g  0  , the steady state values are
independent of  , the relative weight attributed to habit in the utility. In this case (16c) implies
that c  h , so that the steady state level of consumption coincides with the steady state reference
stock level. In that case (16a) reduces to the standard modified golden rule stock of capital,
consistent with the early result of Ryder and Heal (1973), and more recently Alvarez-Cuadrado
et al.(2004).
Third in the presence of positive productivity growth, g , the steady state values are no
longer independent of the relative weight attributed to habit in the utility,  .As shown in
Wendner (2011) this effect occurs because the elasticity of marginal utility becomes a channel
through which a consumption externality affects the steady state equilibrium —even in the
absence of elastic labor supply and a consumption–labor tradeoff.
Of particular interest to our model is the effect of habit on the steady state level of capital
and even more importantly the rate of utilization (similarly rate of depreciation). Using
comparative statics we can show that in the presence of non-satiation, 0    1 , and restricting
  1,
k
 0;

u
 0;


0

meaning that introducing habit formation into the utility function unambiguously increases the
equilibrium normalized stock of capital while decreasing the equilibrium rate of capital
utilization and depreciation. This is a result of the smoother adjust in consumption that takes
place in the presence of a reference benchmark for consumption. An increase in the weight of the
reference benchmark requires an increase in capital to maintain the future value of consumption.
This positive effect of habits on capital accumulation is a result of the fact that in the
presence of positive productivity c  h , and the only way to maintain this relationship is by
having a higher steady state stock of capital. In other words, given that shocks to consumption
are bad, having a higher steady state capital stock hedges against the deleterious effect of a
negative production shock. Furthermore, given a fixed level of output and increasing importance
of habits, the reference consumption level magnifies the effect of a negative consumption shock,
and thus more steady state capital level must accrue to adequately mitigate this “habit risk.”
Finally, we look briefly at the effects of the elasticity of depreciation with respect to the
rate of capital utilization,  , on the steady state values of the rate of depreciation (  ), and on the
rate of utilization  u  . An increase in  , decreases the steady state rate of depreciation. Regarding
its effect on the steady state rate of utilization, there is no loss of generality, and the analysis is
simplified, if we assume g  n  0 . With this assumption the steady state rate of capital
1
 

utilization reduces to u  
 . According to this expression full utilization of capital,
 d   1 
i.e. u  1 , happens when   1   d 4 and when    . Given that u is between zero and one
we can say there is a “U relationship” between u and  . In other words, there is an inflection
point, say  inf , for which the effect of  on u is negative to the left of that point,    inf , and
positive to the right of that point,    inf 5.
To consider the transitional dynamics linearize (18) around the steady state. The
dynamics can be approximated by the third-order system, presented below:
k  

  
0
h   
  
1   2
   B k 
Ch
  Ch  1  g
0
C   k  t   k 


 C   h  t   h 


0    t    


(21)
Economic intuition have us believe that with k and h being sluggish variables while  is
free to jump instantaneously, the system should have a unique stable adjustment path (i.e. saddle
path stability) and thus have two negative and one positive eigenvalues. In what follows, we
discuss only the results and relegate the corresponding derivations to the appendix. It can be
4
In page 2013 of Chatterjee (2005) it reads”…as  increases above 1  d  , u falls below 1…”.There is a typo in
this statement; the “cut-off value” is 1   d and not as presented in the paper.
5
For a detailed analytical discussion see Chatterjee (2005)
easily verified that the sign of the determinant of the matrix (21) is positive. This, however, is
consistent with having either two negative and one positive root, or three positive roots. To
ensure the system exhibits saddlepoint behavior requires extra conditions, but in all of our
simulations, however, we find that (21) exhibits saddlepoint behavior and we focus our attention
on that case, it being the plausible one.
4. Macroeconomic Equilibrium: centrally planned economy
In deriving this optimum, the individual agent neglects the externalities present in
consumption. As a consequence, the macroeconomic equilibrium generated by the decentralized
economy may diverge from the social optimum. In the context of consumption externalities this
has been analyzed by Turnovsky and Monteiro (2008) and Liu at al.(2004). The central planner
understands that consumption reference stock depends upon the economy-wide average
consumption level, [which equals the consumption of the representative agent], and thus
internalizes the impact of the agent’s current consumption decision on the future evolution of the
reference stock, in accordance with (3). Performing the maximization, the optimality conditions
become.
The optimality conditions for this problem are,
U c    
1      uk 


(22a)
 d  u 1
(22b)

 1    u1 k   du   n  g 

U

   h    g


(22c)
(22d)
where  denotes the agent’s shadow value of capital,  is the shadow value of the agent’s
reference stock, together with the transversality conditions:
lim e
t 
      11  g  t
  t  k  t   lim e
t 
      11  g  t
 t  h t   0
(22e)
The interpretation of the optimality conditions (22a)-(22e) has some important
differences from the decentralized economy. First equation (22a) equates the utility of an
additional unit of consumption adjusted by its impact on the future reference stock to the shadow
value of capital, while equation (22b) determines the optimal rate of capital utilization by
equating the marginal benefit to marginal cost. Equation (22c) is the standard intertemporal
allocation condition equating the social rate of return to capital to the social of return on
consumption. Equation (22d) is an additional intertemporal allocation condition equating the
marginal disutility of an additional unit of the reference stock measured in terms of its shadow
value to the cost of the reference stock.
From equation (22.a) we obtain6
c  C ( h,  ,  )
And we define like before
Ch  
U ch
;
U cc
(23a)
C 
1

 0, C  
0
U cc
U cc
(23b)
Using (23a) the macroeconomic equilibrium is terms of the canonical for the central planner is
described by
1    1    

k   B1 
 k  c  h,  ,     n  g  k



(24a)
h    c(h,  ,  )  h   gh
(24b)

   1  1  1 
 B1 k  
  

    n  g   1    
(24c)
      g     U h
(24d)

6
Where U ij denote the second order partial derivatives of the utility function with respect to the two arguments For
simplicity we use the utility notation instead of the actual results.
1
  1       1
where B1  

 d 
Imposing the stationary conditions, k  h      0 on the system (24a)-(24d) we can
determine the central planner's optima for k , c, h and  in a similar recursive manner as we did in
(19a)-(19c). Letting * denote the optimum values; we summarize these expressions as follows:
  1     1 B11 
*
k 

   g  n  

  1

(24a)
 1     * 
*
c*   B11 1 
k   n  g  k
 

h* 

g
c* 
c*   g

h*

(24b)
(24c)
U h  c* , h*  

 
     g 
(24d)
 *  U c  c* , h*    *
(24e)

In addition, using the steady state value of capital (24a) we can derive the steady state
values for the rate of capital utilization and the rate of depreciation
1
   n     1      g  
u*  

d   1


(25a)
and
   n     1      g 

  1


*  
(25b)
The parallels between these optimum values and (20a)–(20b) for the decentralized
economy are clear. Indeed if we eliminate the tax rates, all the expressions remain unchanged.
This is not surprising in light of the results obtained in Liu et al.(2004), and Turnovsky et
al.(2008), who show that consumption externalities have no effect in the steady state if labor
supply is exogenous.
To consider the transitional dynamics linearize (23) around the steady state. The
dynamics can be approximated by the fourth-order system, presented below:

k  
  
0
h   
   B 1  k *   2  *
1
  

   
0
Ch
C
  Ch  1  g
 C
0
0
 U hh  U hcCh  U hcC
C   k  t   k * 


 C   h  t   h* 


0    t    * 

a44     t    * 
(26)
with a44    g    U hcC
Economic intuition have us believe that with k and h being sluggish variables while 
and  are free to jump instantaneously, the system should have a unique stable adjustment path
(i.e. saddle path stable) and thus have two negative and two positive eigenvalues. In what
follows we discuss only the results and relegate the corresponding derivations to the appendix. It
can be easily verified that the sign of the determinant of the matrix (18) is positive. This,
however, is consistent with there being either two negative and two positive roots, four positive,
or four negative roots. To ensure the system exhibits saddlepoint behavior requires extra
conditions, which turn out to be very simple and weak in nature; namely that the diminishing
marginal utilities dominate, i.e.
U cc  U hc  0 , U hh  U hc  0
5. Optimal Tax Policy
The objective is to characterize a tax structure such that the decentralized economy
mimics the dynamic equilibrium path of the centrally planned economy. In the standard
approach, there is no qualitative difference between taxing only the gross rental capital
income or to tax rental income net of depreciation. In both cases the optimal policy prescribes
zero taxation in the long-run [(Chamley (1986) and Judd, (1985)]. But once capital utilization
decisions are endogenized, the optimal taxation policy is sensitive to the way we define
taxable income.
It is important to remember that, as was shown by Liu et al. (2005), and Turnovsky et al.
(2008), in the absence of endogenous labor the long-run solution for the decentralized and
the central planner coincide, i.e. the consumption externality does not affect the steady state
only the transition. Therefore, it is not surprising that the optimal tax discussion presented
here does not depend on the consumption references parameters. In contrast, endogenous
utilization decision will play a key role in the determination of the optimal tax policy.
For a better comparison with the existing literature it is convenient to consider two
possible scenarios. In the first case, the government only levies the income tax, ys because
firms are not allowed to deduct capital depreciation. In light of the previous discussion the
government taxes income gross of capital depreciation. However, if you read the IRS
publication 946, titled how to depreciate properly, we see that real-world tax systems
consider some degree of depreciation deductions for tax purposes. Therefore, in the second
scenario capital depreciation allowances are possible.
An immediate difference between the standard Ramsey model and a model with
endogenous utilization is that in the Ramsey model we only need to worry about the effect of
taxation on the intertemporal decisions of capital accumulation. In contrast, the
endogenization of capital utilization, adds another source of distortion to the agent decisions.
Now, and in addition to the intertemporal distortion, the agent must also worry about the
static allocation (intratemporal decisions about utilization ) distortion arising from the choice
of the rate of utilization.
In sum, in the presence of endogenous utilization we must consider not only the long-run
analysis, (intertemporal effect on capital accumulation decision), but also the short-run or
static effect on capital utilization decision.
i)
Optimal income tax in the absence of depreciation deductions.
Proposition 1: in the absence of capital depreciation allowances,  d  0 , the optimal
long-run tax rate on capital is zero, ˆys  0 . Therefore, we can say the Chamley-Judd
result of zero long-run optimal income tax, holds in the presence of endogenous
utilization decisions.
Proof: looking at expression (19a) and (24a), and setting  d =0, we can see that
k  k * the result in (24a)
  1     1 B11 
*
k 

   g  n 


  1

1
  1       1
with B1  

 d 
can be reproduced by the decentralized case (19a)
  1     1 1  
k 
   g  n 

s
y
B
1



  1

 1     1    

with B  


d


s
y
1
  1
If and only if  ys  0
In the end, this is to be expected because the endogenization of the utilization decision
does not affect the intertemporal margin of capital taxation.
Proposition 2: this result of zero long-run income taxation also extends to the short-run
income taxation, i.e.  ys  0 for all periods of time.
Proof: the optimality condition given by expression (22b) implies that
1      uk 

 d  u 1
Whereas the intratemporal condition of the decentralized economy, expression (10b)
[combined with (13a)], yields
1   1    uk 
s
y

 d u 1
Replication of the central planner optimal value requires that  ys  0 in the short-run too.
The explanation for this result relies on the fact that in the presence of
endogenous utilization, capital supply is perfectly elastic not only in in the long-run (via
investment decisions) but on the short-run as well (via utilization decisions.) Therefore,
income taxation produces both a long-run as well as a short-run distortion, which are so
large in terms of utility that the optimal tax policy requires  ys  0 in both the long and
the short-run.
ii)
Optimal tax policy in the presence of depreciation deductions.
The previous result is based on the idea that government taxes capital gross of
depreciation. In contrast to the standard neoclassical model where the distinction between
gross and net is inconsistent for optimal tax purposes, we can show that the same is not
true when utilization decision is endogeneized. In this case the intratemporal utilization
decisions create a departure from the traditional optimal tax policy analysis.
Proposition 3: in the presence of deductible capital allowances , the optimal long-run
taxes should be equal to zero, i.e.  ys   d  0 . Therefore the Chamley-Judd results of
zero long-run income taxation still holds in the presence of depreciation allowances.
Proof: this proof is similar to the one for proposition 1 and the only difference is that
now we compare directly expression (19a) and (24a), without assuming that  d  0 .
Proposition 4: The result of zero long-run income taxation does not extend to the shortrun, and to eliminate the short-run distortion the optimal tax rate equals  ys   d i.e. we
must have what is known as full expensing.
Proof: combining expression (10b)with (13b) (reproduced here for convenience),yields,
1   1    uk 
s
y
and combining with (22b) we get

 1  d  d u 1 (10b’)
 ys 1      uk 

  d d  u 1
(27)
which requires proposition 4, that depreciation allowances equals the rate of income
taxation.
In the presence of endogenous capital utilization expression (27) and the ensuing
there should be no wedge between the marginal user cost of capital and the marginal
product of capital services.
In other words, looking at expression (10b’), it means that the after tax marginal
rate of return of investing in capital must equal the after rebate marginal cost of using
capital. In this case the government distorts the intratemporal margin of capital utilization
via two different channels:
 ys
i)
by taxing income at rate
ii)
rebating the depreciation of capital utilization at rate
d
In sum, we saw that in the absence of tax allowances for capital utilization, the ChamleyJudd result of zero optimal income tax, holds for both the short and th long run. In contrast if
agents are allowed to use full expensing (i.e. fully deduct capital from rental capital income),
then the optimal income tax rate is different from zero in the short-run.
It is important to notice that the result of non-zero optimal short-run taxation, only holds
in the case of full expensing. If agents were only allowed to deduct a fraction of the depreciation,
then the zero short-run optimal income tax would still hold. Intuitively, imagine the depreciation
has two components: normal and abnormal7. Assume, for simplicity, that the normal component
is the typical constant depreciation rate, and that the abnormal is the depreciation resulting from
the intensive utilization. Then if agents can only deduct the normal component, it will be
resemble the standard Ramsey case and thus he zero optimal short-run income tax will result.
7
For instance this expression could be expressed as  u    N   u   N  , with the second term representing the
abnormal component.
6. Numerical Analysis
To get a better understanding of the implications of combining endogenous capital
utilization and habit formation, we calibrate the model and perform some dynamic analysis.
Before we proceed it is worth mentioning that, in light of the dynamic analysis performed in
Alvarez-Cuadrado et al. (2004), the absence of endogenous labor makes both the external
(decentralized solution) and the internal case very similar. Therefore, we ignore the external case
and contrast the dynamic behavior of our specification under the habit formation case8, with
three other economy specifications that have been widely used in the literature. These models
can be seen as special cases of our model which are obtained by considering polar cases of the
parameters  and  , namely: (i) neoclassical growth model9 (   0,    ), (ii) the habit
formation case (    ) and (iii) the capital utilization case (   0 ).
Table 1 summarizes the parameters upon which our simulations are based and most of
these are standard and non-controversial. In this regard,   0.65 , the rate of time preference
  0.03 , the instantaneous intertemporal elasticity of substitution , 1   0.4 , population
growth rate n  0.015 and the growth rate of labor productivity  g  of 2 percent are well
documented, and this being a neoclassical model, normalization   1 is unimportant.
[INSERT TABLE 1 AROUND HERE]
On the preference side, the critical parameters pertain to the relative importance of the
reference stock,  , the speed with which it is adjusted,  . In this regard, we follow AlvarezCuadrado et al. (2004) very closely, thus making it easy to analyze the implications of extending
the habit formation model to account for endogenous capital utilization decisions. Therefore, we
set   0.5,   0.2 as benchmark values. However, since the information on these parameters is
8
For this section we solve the habit formation case using a budget constraint of the type
k  1   ys  uk 
1
 1   c  c   n  g  1   d   u   k . The solution to this problem is straightforward but is
available from the authors on request. It is left out here to save space.
9
Setting   0 the reference stock, H, is irrelevant for utility, in addition when    we have full utilization of
capital u  1
sparse, we let  vary to 0 (conventional case), 0.2, 0.8 (this later case based on the estimates
provided by Fuhrer (2000)) and 1.
On the production side, different estimates for the elasticity of depreciation with respect
to the rate of capital utilization,  , can be found in Burnside and Eichenbaum (1996), Finn(1995)
and Dalgaard (2003). We consider the benchmark value to be   1.7 , which lies in the middle of
its estimated range of 1.4-2. However, in light of the different estimates we let  equal 1.4, 1.45,
1.5, 1.56, 1.6 and 2. In this regard, we follow Chatterjee (2005) very closely, thus making it easy
to analyze the implications of extending the capital utilization decision model to account for
habit formation. Given the lack of empirical estimates for d we set it at 0.3, by following the
discussion in Chatterjee’s (2005) footnote 19.
The calibration exercise is as follows: for any given value of  the model derives an
optimum depreciation rate (  ), and we use this value to calibrate the neoclassical and the habit
formation growth models. We then compare the speeds of convergence and other relevant
equilibrium quantities from the models by varying  and  .
We begin this section by contrasting the steady state of the different models, and then
focus on the dynamic response to two fiscal shocks, namely: i) a 10 percent increase in income
tax, from 0 to 10% and ii) a 10 percent increase in the rate of tax deduction for depreciation.
6.1. The Steady State Equilibrium.
The goal of this section is to contrast the steady state values of consumption and output in
different models with our model. This analysis is done assuming the taxes are zero, and since the
steady state of these variables does not depend on  , we focus on the effects of  and  . Table
2A-2D reports relative differences in scale-adjusted output and consumption between our model,
the habit formation model, the neoclassical model, and the capital utilization model.
The relative differences in scale-adjusted quantities are defined as 1  Z Z i  , where Z
is the equilibrium value for the model of capital utilization with habit formation and Z i
corresponds to the value of one of the other three models, namely the habit formation, the
neoclassical and the capital utilization.
The first thing to notice is that the neoclassical and the habit formation models constantly
overstate the steady state equilibrium of output and thus consumption, whereas the capital
utilization model always understates the steady state equilibrium of output and consumption. The
overall underestimation of the level of output in the capital utilization spans the range of 1 to
10.78 percent for output and 0.6 and 6.42 percent for consumption. This underestimation is
increasing in  and  . For example, when   0.8 [as estimated by Fuhrer (2000)], using the one
sector capital utilization model underestimates the value of output between 4.558 and 8.16
percent.
[INSERT TABLE 2 AROUND HERE]
In the case of habit formation, the differences are the same for output and consumption
and range between 10.86, when   0.2 , and 33.75 percent, when   1 . If one were to take the
value of   0.5 as used in Alvarez-Cuadrado et al (2004), we see that the steady state is
overstated between 14.44 and 29.94 percent, which is quite a significant value. In the
neoclassical case the overstatement is smaller than in the habit case ranging between 9.94 and 26
percent for output, and 10 and 30 percent for consumption.
In the limit when    , capital utilization converges to 1, and depreciation becomes
zero, two “groups” are formed. On the one hand we have the “habit formation group”, because
our model and the habit formation model converge to the same steady state level of output. On
the other hand, we have the “one sector group” with the capital utilization and the neoclassical
growth model converging to the same level of output. In this case the “one sector group”
underestimates the “habit formation group” by values that range between 2.5 and 14.2 percent,
while output underestimates range between 3.5 and 22.6 percent.
[INSERT TABLE 3 AROUND HERE]
Table 3 looks at the implications of habit formation on the rate of capital utilization and
depreciation. The immediate conclusion is the presence of a negative relation between the
optimal rate of capital utilization as well as depreciation and the weight of habits in utility,  .
Additionally and for the range of  used in our numerical simulations depreciation and the rate
of capital utilization are decreasing in  10.
According to the first row of Table 3, when   0 , we see that the optimal capital
utilization in the no habit model varies between 84 percent and 56 percent. Introducing habit
formation into this model reduces capital utilization by significant amounts, with the values
ranging from approximately 4 percent, when   0.2 and   1.4 , and 31 percent when   1 and
  1.4 .
In sum, Table 2 and 3 highlight the fact that failure to account for the joint effect of two
empirical relevant effects, capital utilization and habit formation, may lead to significant
misestimation of equilibrium quantities.
6.2.Transitional dynamics.
This section compares the implications of combining the optimal decision of capital
utilization with habit formation with those obtained under a model of just capital utilization. We
do this by looking at the dynamic adjustment of key variables after: i) a 10 percent increase in
income tax, from 0 to 10% and ii) a 10 percent increase in the rate of tax deduction for
depreciation. To illustrate the key differences we focus on the adjustment of three variables:
consumption, the rate of capital utilization and capital.
[INSERT TABLE 4A AROUND HERE]
Table 4A, summarizes the steady state values of consumption, capital and rate of
utilization for the benchmark parameters defined in table 1. According to Table 4A, we see that
in a model of capital utilization the steady state values of consumption and capital are only 97.6
and 81.1 percent of the steady state in the model of capital utilization with habit formation. As a
result of the underestimation of capital, the former model requires a higher rate of capital
utilization which is about 10 percent higher than in the latter. Hence, ignoring the combination of
endogenous capital utilization decisions and habit formation in consumption implies the model
underestimates consumption and capital, while overestimating the rate of capital utilization.
10
The U relationship mentioned during the model derivations occurs for values of   2
It is important to notice that since the model of capital utilization and our model do not
have the same steady state, and thus it is convenient to normalize the results so the graphical
interpretation is made easier. Therefore, we plot the variables as a fraction of their initial
equilibria. After a shock, the economies converge to a steady state value which can easily be
converted into a percentage reflecting how much the new steady state differs from the initial
steady state.
6.2.1. 10 percent increase in income tax
The first thing to notice is that this type of shock does not have a permanent effect on the
optimal rate of capital utilization, see expression (20a)and (25a), but affects the transitional
adjustment after the shock, see expression (14).
[INSERT TABLE 4B AROUND HERE]
From Table 4B, we see that a 10% permanent increase in  ys reduces the steady state value
of capital by 15% percent in all the models. Consumption, on the other hand, is reduced in the
long-run by different amounts for all the models; with the magnitudes reflecting the different
forces at play in each model. To better understand the implication of this policy one must
consider the short-run impact, because the introduction of endogenous utilization makes capital
supply perfectly elastic to the tax increase.
In the short-run, however, table 4B illustrates that the combination of habit formation in
consumption with capital utilization inhibits the jump in consumption at the time of the shock
relative to the models without utilization, and increases it relative to the model without habits. In
the models with endogenous utilization, the rise in consumption is accompanied by a fall in the
rate of capital utilization of 7.5% .
Intuitively, in the absence of utilization decisions, the response to the rise in income tax,
must be met exclusively by a rise in consumption; and thus consumption increases by 4.6 and 4.4
percent in the model of habit formation and the Ramsey model respectively. In the presence of
utilization decisions, firms immediately reduce utilization and thus consumption rises by less.
The combination of demand (habit) and supply (utilization) forces leads to a smaller rise in
consumption of 1.25%, which is larger than the 0.2% rise in the case of utilization but no habits.
Therefore, as capital accumulation decreases over time, so does consumption, but the small
initial gain in the presence of utilization leads to a bigger long-run loss in consumption, relative
to the models without utilization decisions.
6.2.2. 10 percent increase in rate of tax deductions for depreciation
As shown analytically an increase in the tax allowances for depreciation raises utilization
in the long-run. The interesting fact is the overshooting observed in the short-run, with utilization
rising on impact by 8.12% against the 6.39% observed in long-run. Intuitively, this is can be seen
as a way to accommodate the fact that capital accumulation cannot change in the short-run, thus
the answer must come through the service provided. In the long run, capital begins to increase
and firms can reduce the utilization, but the tax deductions still leave utilization above the initial
value.
It is also interesting to notice that the only difference in response between the model of
capital utilization and capital utilization with habits, is the short-run impact on consumption.
This is a result of the way we introduced the government in the model, that eliminates the
depreciation allowance from all parts of the model except B and steady state utilization. In other
words, it closely resembles a productivity increase.
7. Conclusion
This paper adds endogenous capital utilization decisions and habit formation in
consumption into a neoclassical growth model, in order to bring together two empirically
relevant components. Our objective is to investigate the tax policy implications of this
combination for the steady state, and the transitional dynamics.
First, we derive the analytical results for our model and show that in the presence of labor
productivity the combination of capital utilization and habit formation reduces the optimal value
of capital utilization. On the other hand, in the absence of labor productivity, habits have no
effect on capital utilization.
Second, our optimal tax results show the Chamley-Judd result of zero long-run optimal
income tax, holds in the presence of endogenous utilization decisions, irrespective of looking at
the net of gross return on capital. The result, however, only extends to the short-run if there are
no depreciation allowances.
Our simulations on simple tax policy changes show important differences at different
levels between our model and the capital utilization, the neoclassical and the habit formation
models. Our results suggest that by ignoring the combination of endogenous capital utilization
decisions and habit formation, economic growth models may be understating the magnitude of
long-run equilibrium in the case of the capital utilization model, but overstating it in the case of
habit formation and the neoclassical model. In addition, this simple tax exercise shows that there
are significant differences in combining demand and supply forces. It would be interesting to
consider the analysis in a richer environment of, for instance endogenous labor supply or
productive government spending
In sum, our results reinforce the idea that research on economic growth should pay more
attention to the interaction of these two models as a way to analyze the implications of economic
policy. Thus, in the future it would be interesting to investigate the consequences of
endogenizing labor and analyzing fiscal policy implications. Under this setting, it would also be
interesting to consider the potential effects on dynamic scoring
Appendix
We may write the characteristic equation to (18), namely
   1      1 g

0

1
  B  k   2 

0

in the form
Ch
  Ch  1  g
C
 C
0
0
 U hh  U hcCh  U hcC
C
 C


0

0

  1      1 g  g    U hcC 
(A1)
 4  1 3   2 2  3   4  0
(A2)
where the solutions,  i , i  1,..., 4 , are the eigenvalues. Note that
 1 2 3 4  4 (determinant)
(A3)
1  2  3  4  1 (trace)
(A4)
It can be verified that
4 =  B1 k  2 (   g )(     1    g   )  U cc  0
(A5)
and that
1   1  2  3  4  f k   (Ch  1)  g    1     1 g  g    U hcC
 f k    1      1 g
(A6)
 2  1      1 g  0
Result (A5) is consistent with there being either two negative and two positive roots, four
positive or four negative roots. Whereas, result (A6) implies that not all eigenvalues are negative.
In addition, Descartes rule of signs in coefficients of (A2) determine the number of positive
(unstable) roots.
Denote the determinant in (A1) by D say,
  1      1 g 
D
0
 B1 k  2
0
Ch
  Ch  1  g 
C
 C
C
 C
0
 U hh  U hcCh 

U hcC
0
  1      1 g  g    U hcC 
to simplify the notation we define     1     1 g
Adding row two to row one after multiplying it by 1  , yields
 
D
0
 B1 k   2
0
 g  
 1 
 

  Ch  1  g 
0
 U hh  U hcCh 
  Ch  1  g 
 (  v)
0
 U hh  U hcCh 
 C

U hcC
 g  
 1 
 

 B1 k   2   Ch  1  g 
 U hh  U hcCh 
0
0
 C
 C

0
U hcC   g   
 C
0
 D1
  g   
0
 C
U hcC
0
 C
 D2
  g   
Now evaluate D1 :
D1   (  ) (  U hcC  )   (Ch  1)  g  v    C [U hh  U hcCh ]
For notational simplicity let
U hhU cc  U hc2
U  U hh  U hcCh 
0
U cc
Hence
D1   (  )  2   (Ch  1)  g   g    U hcC     (Ch  1)  g  [  g    U hcC ]   CU 
  (  )  (  )    (Ch  1)  g      (Ch  1)  g   
  2 (  ) 2  (v  )   (Ch  1)  g  [   (Ch  1)  g ]   CU 


D1   4  2 v3   2    (Ch  1)  g  [   (Ch  1)  g ]   CU  v 2
   (Ch  1)  g  [   (Ch  1)  g ]   CU  v
Now evaluate D2 :
D2   B1 k  2    g   C   g  U hcC  v   CU hcC 
  B1 k  2    g   C   g    v 
  B1 k  2 C  v 2  v    g    (    ) 
Thus, combining terms yields the characteristic equation
D   4  2 v3   2v 2  3v 
 B1 k  2
U cc
(  g   )  g     0
(A7)
where
 2   2     Ch  1  g  [    Ch  1  g ]   CU  
 B1 k  2
3       Ch  1  g  [    Ch  1  g ]   CU  
 B1 k  2
U cc
U cc
 2
  
In order to have four positive eigenvalues, Descartes rule of sign would require four sign changes
and hence 2  0,3  0 . But we can rule this out with mild restrictions. For this sign
pattern would require    2  0,   0 . But under weak conditions,   0 . Evaluating  , we
have
     Ch  1  g      Ch  1  g    CU  
 B1 k  2
U cc
2
  U  U hc 
   U cc  U hc 

 B1 k  2
2 U 
     cc

g



g











U cc
 U cc 
  U cc 
   U cc 

Thus yielding the weak condition presented in the text. We therefore have a unique stable
equilibrium path, and the solution to the system is of the form,
k (t )  k  B1e1t  B2e 2t
h(t )  h  B121e1t  B222e 2t
 (t )    B131e t  B232 e t
1
2
 (t )    B141e t  B242e t
1
2
where B1 , B2 are arbitrary, k (0)  k0 , h(0)  h0 , and the vector (1, 2i , 3i , 4i ), i  1, 2 is the
normalized eigenvector associated with the stable eigenvalue  i . The constants B1 , B2 are
obtained by initial conditions namely
B1  B2  k0  k
21B1  22 B2  h0  h
and will depend upon the shock
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